In an isosceles triangle ABC, the vertex A is (6,1) and the equation of the base BC is 2 x + y =4. Let the point B lie on the line x+3 y=7. If (α, β) is the centroid triangle A B C, then 15(α+β) is equal to :

Question

In an isosceles triangle ABC, the vertex A is (6,1) and the equation of the base BC is 2 x + y =4. Let the point B lie on the line x+3 y=7. If (α, β) is the centroid triangle A B C, then 15(α+β) is equal to : (A) 39 (B) 41 (C) 51 (D) 63

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/c9528b983480fff4554f0557572c0ada-.png" /> Point B (1,2) Now let C be (h, 4 - 2h) (As C lies on 2 x + y =4 ) ∵ Δ is isosceles with base BC ∴ A B=A C √25+1=√(6-h)2+(2 h-3)2 √26=√36+h2-12 h+4 h2+9-12 h 26=5 h2-24 h+45 ⇒ 5 h2-24 h+19=0 ⇒ 5 h2-5 h-19 h+19=0 h=(19/5) or h =1 Thus C((19/5), (-18/5)) Centroid ((6+1+(19/5)/3), (1+2-(18/5)/3)) ((35+19/15), (15-18/15)) ((54/15), (-3/15)) α=(54/15) ; β=(-3/15) 15(α+β)=51

Q. In an isosceles triangle ABC, the vertex A is (6,1) and the equation of the base BC is 2x+y=4. Let the point B lie on the line x+3y=7. If (α,β) is the centroid △ABC, then 15(α+β) is equal to :

39

41

51

63

Q. In an isosceles triangle $A BC$ , the vertex $A$ is $(6, 1)$ and the equation of the base $BC$ is $2 x + y = 4$ . Let the point $B$ lie on the line $x + 3 y = 7$ . If $(α, β)$ is the centroid $△ A BC$ , then $15 (α + β)$ is equal to :